How 251 is a prime number?

Answer 1

One way is to test all the prime numbers equal or less than Sqrt[251]. These are: {2,3,5,7,11,13}.

We can eliminate 2, 3, and 5 because the number isn't odd, it's digits don't sum to a multiple of 3, and the last digit isn't a 5 or a 0.

For 7, note that 252 = 7*36, so 7 cannot divide 251 because that would imply 7 divides 1.

For 11, note that 253 = 11*23, so 11 cannot divide 251 because that would imply 11 divides 2.

For 13, note that 13*19 = 247, so 13 cannot divide 251 because that would imply 13 divides 4.

Hence, 251 is prime